Fourier Transform

Historical Aspects

The Fourier transform is named after the French engineer Jean Baptiste (Joseph Baron) Fourier (1768–1830), who, motivated by his work modeling heat conduction, proposed that any function could be decomposed into a superposition of sinusoidal (sine and cosine) terms. It has since been found that the decomposition is valid only for functions that satisfy certain conditions, which are rather technical (the interested reader is referred to the references). However, almost all functions that arise in physical applications will satisfy the conditions or at least will be well approximated by a sum of sinusoidal terms.

Which Fourier Transform?

Without context, the term Fourier transform generally refers to the continuous Fourier transform, which is a linear mapping from a continuous time interval to the frequency domain. The discrete Fourier transform is then the equivalent form for discrete time.

The frequency domain representation allows analysis of a function's frequency characteristics, such as the contribution of the function to sinusoids at different frequencies. The inverse Fourier transform reverses the transformation from the frequency domain back to the time domain. The mapping is unique, so the inverse Fourier transform will reproduce the original function exactly. The dual transforms are called the Fourier transform pair.

What Do We Mean by Frequency?

Frequency is the number of times a function repeats itself within a unit of time. If the time unit is a second, the frequency measure is hertz, the number of cycles per second. The period is the time taken for the function to repeat; in other words, period is the reciprocal of frequency: Frequency = 1/Period. Frequency can intuitively be considered in terms of sound waves: A bass note is a low-frequency sound, and a whistle is a high-frequency sound.

The period between successive peaks of the solid line in Figure 1 is 8 seconds, giving a frequency of f = ⅛ Hz. The dotted line is of a higher frequency, f = ¼, shown by the period of 4 seconds. The amplitude of a [Page 366] sinusoid is the height of each peak or trough. The solid line has amplitude one, whereas the dotted line has amplitude ¼. The phase of a sinusoid indicates which part of the cycle the function commences at time zero. The solid line has phase 0 (a sine term), whereas the dashed line has the same frequency, but the phase is shifted by π/2 (equivalent to a cosine term). Hence, cosine functions are simply phase-shifted sine functions.

Figure 1 Sinusoidal Functions of Various Frequencies and Phases

Note: Solid line is sin(2π ft) with frequency f = ⅛, dashed line is sin(2π t + π/2) = cos(2π ft), and dotted line is sin(2π ft)/4 with twice the frequency, or f = ¼.

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